Group decision-making model using fuzzy multiple attributes analysis for the evaluation of advanced manufacturing technology

Fuzzy Sets and Systems 160 (2009) 586 – 602 www.elsevier.com/locate/fss

Group decision-making model using fuzzy multiple attributes analysis for the evaluation of advanced manufacturing technology Shian-Jong Chuu∗ Department of Business Administration, Nanya Institute of Technology, 414, Sec. 3, Chung-Shang E. Road, Chungli, Taoyuan 320, Taiwan, ROC Received 24 November 2007; received in revised form 24 July 2008; accepted 29 July 2008 Available online 19 August 2008

Abstract Selection of advanced manufacturing technology is important for improving manufacturing system competitiveness. This study builds a group decision-making model using fuzzy multiple attributes analysis to evaluate the suitability of manufacturing technology. Since numerous attributes have been considered in evaluating the manufacturing technology suitability, most information available in this stage is subjective and imprecise, and fuzzy sets theory provides a mathematical framework for modeling imprecision and vagueness. The proposed approach involved developing a fusion method of fuzzy information, which was assessed using both linguistic and numerical scales. In addition, an interactive decision analysis is developed to make a consistent decision. When evaluating the suitability of manufacturing technology, it may be necessary to improve upon the technology, and naturally advanced manufacturing technology is seen as the best direction for improvement. The flexible manufacturing system adopted in the Taiwanese bicycle industry is used in this study to illustrate the computational process of the proposed method. The results of this study are more objective and unbiased, owing to being generated by a group of decision-makers. © 2008 Elsevier B.V. All rights reserved. Keywords: Fuzzy sets; Multiple attributes analysis; Group decision-making; Advanced manufacturing technology; MEOWA operators

1. Introduction Manufacturing environments recently have changed so fast that manufacturing system competitiveness has increased importance. Manufacturing firms have been investing in advanced manufacturing technologies (AMTs) to improve their manufacturing performance in terms of cost, quality, and flexibility, in an effort to compete with other firms in the global marketplace [20]. Generally, AMT represents numerous modern technologies devoted to improving manufacturing firm competitive position, including flexible manufacturing systems, computer-integrated manufacturing systems, just-intime systems, and so on [32]. Selecting a suitable AMT is important for manufacturing firms when making capital investment decisions to improve their manufacturing performance. In practice, while some firms that adopt these technologies report reaping considerable benefits, others have been less successful, which indicates that AMT investment remains promising but highly risky [32]. Furthermore, the rapid growth of the AMT industry is now creating problems. Prospective firms now face the situation of having to decide among several AMTs, all of which are capable of performing a specific task. The development of appropriate ∗ Tel.: +886 3 4361070x5616; fax: +886 3 4373959.

E-mail address: [email protected] (S.-J. Chuu). 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.07.015

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assessment approaches is crucial to ensuring that each AMT project is assessed from the perspective of all benefits and costs [23,32]. The literature review has revealed difficulties in justification of the AMT investment using traditional economic technology, and a few existing methodologies have provided satisfactory solutions. Therefore, this paper presents a methodology that can be applied to the AMT evaluation problem. From a methodological perspective, the AMT selection problem is a fuzzy multiple attribute and group decisionmaking problem involving the consideration of fuzzy assessments and the opinions of multiple decision-makers or experts. In the AMT selection decision problem, numerous effects (attributes) must be considered in justifying a particular manufacturing technology [20]. Punniyamoorthy and Ragavan [30] proposed a deterministic approach to decision making for technology selection, which considers both subjective and objective attributes, attributes that are also seen in Karsak and Tolga [20]. Objective attributes are defined by using appropriate numerical terms, such as investment cost, setup time, and so on, used for assessing the quantitative effects of manufacturing technology. Since precise quantitative information may not be available or computational costs may be excessively high, these expert identified performance ratings can be ‘approximate numerical values’, which can be expressed by sentences such as ‘approximately equal to’, ‘at least’, or ‘approximately between’, and so on [24,28,29]. Subjective attributes have qualitative definitions, e.g., process flexibility, product quality, etc., used for assessing the qualitative effects of manufacturing technology, which may be unquantifiable due to the nature of such technology. Expert opinions can be represented linguistically, using labels such as ‘high’, ‘middle’, or ‘low’, etc. Consequently, with respect to multiple attribute analysis, appropriate AMT selection is difficult to synthesize. To improve the quality of decisions in fuzzy environments, contemporary organizations prefer group decision-making. Obviously much real world knowledge is fuzzy rather than precise. In AMT ranking/selection problems, assessment data employed in multiple attribute decision-making (MADM) problems are generally fuzzy linguistic, numerical, or some mixture of thereof. Hence, a useful decision-making model is to provide the ability to handle multiple fuzzy assessments, that is, by aggregating the opinions of multiple experts. This study attempts to establish a useful group decision-making model by using fuzzy multiple attributes analysis to improve the AMT selection process. Therefore, this study proposes a group decision-making model based on fuzzy multiple attributes analysis to assess the suitability of AMT alternatives. In the proposed method, we have developed a new fuzzy fusion method of fuzzy information for managing information assessed using both linguistic and numerical scales. Fusion of fuzzy assessment data is performed by maximum entropy ordered weighted averaging (MEOWA) operators. The remainder of this paper is organized as follows. Section 2 provides a literature review focused on methods used in AMT selection, group decision-making with fuzzy multiple attributes analysis, and linguistic assessments used in group decision-making. Section 3 then presents a fuzzy fusion method. A fuzzy multiple attribute group decision-making model for evaluating an appropriate AMT is proposed in Section 4. The process aggregates each parameter assessed by an individual, and aggregates the results to determine the final ranking order. In Section 5, an example using a case of leading Taiwanese bicycle manufacturers is used to illustrate the computational process of the proposed method. Finally, the last section summarizes this research. 2. Literature review This section reviews the literature on justification techniques for AMT, group decision-making with fuzzy multiple attributes analysis, and linguistic assessments used in the group decision-making. 2.1. Justification techniques for AMT Numerous precision-based methods of AMT evaluation have recently been developed. These deterministic justification techniques are classified into economic, analytical, strategic, and integrated approaches [26]. These methods deviate from one another mainly because of the treatment of qualitative effects being based on crisp evaluation, i.e., the evaluation values must be precise. However, in real life, the assessments of performance ratings for subjective attributes (or the importance grades of all attributes) are generally expressed via fuzzy linguistic assessment [39]. Therefore, the primary problem of the above methods is that they are based on accurate measurement and crisp evaluation. Classical methods of solving AMT selection problems cannot effectively handle problems involving imprecise and subjective information. In fact, most decision-makers (or experts) view performance ratings for subjective attributes as linguistic labels, such as high, middle, low, etc. Since Zadeh [38] introduced fuzzy sets theory to deal with vagueness problems, linguistic labels have been used in approximate reasoning within the fuzzy sets theory framework to handle

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imprecise data and vague linguistic expression. Several researchers have utilized fuzzy multiple attributes decisionmaking (FMADM) models for AMT selection problems. Perego and Rangone [29] presented reviews of the application of FMADM theory to AMT selection. FMADM techniques belong to three categories: (1) fuzzy goal methodology, (2) fuzzy linguistic methods, and (3) fuzzy hierarchical models. Karsak and Tolga [20] proposed a FMADM method for evaluating advanced manufacturing system investments. The proposed approach applied fuzzy discounted cash flow analysis and linguistic assessments of decision-makers to the economic (objective) criterion and strategic (subjective) criteria, respectively. Abdel-Kader and Dugdale [1] proposed a FMADM model for the evaluation of AMT investments. The proposed model applied the mathematics of analytical hierarchy process and fuzzy sets theory to aggregate the two major dimensions of financial attribute and non-financial attributes, respectively. Moreover, Punniyamoorthy and Ragavan [30] proposed a strategic decision-making model for AMT selection, which considered objective and subjective factors. Regarding the intangible benefits arising from subjective attributes, a range of evaluated values is taken and three different levels are considered within the range. However, these methods allow the group decision-making scenario to be ignored, which deter management from using FMADM methods. Many researchers have focused on increasing the ability of groups to make quality decisions [7,16]. Ideally, groups should be able to achieve better decisions than individuals because of having greater collective knowledge. Choudhury et al. [7] proposed a multi-criteria, multi-preference group decision-making model for AMT selection. In this method, group decision-makers can express their preferences in four different ways, i.e., utility functions, preference ordering of alternatives, fuzzy preference relation, and multiplicative preference relation [5,13]. This study modeled a technology selection problem defined into a multiple decision-maker and multi-attributes scenario, and proposed a group decisionmaking model with a fuzzy fusion method for resolving manufacturing technology evaluation problems. 2.2. Group decision-making with fuzzy multiple attributes analysis Decision-making is a usual human activity. It basically involves selecting the most preferred alternative(s) from a finite set of alternatives in order to achieve certain predefined objectives (or goals). A literature review of AMT evaluation demonstrates that the AMT selection problem is a multi-attribute group decision-making problem in a fuzzy environment, and involves considering both multiple attributes and group decision-making. The growing complexity and uncertainty of decision situations make it less and less possible for a decision-maker to consider all relevant aspects of a problem, and necessitates the participation of multiple experts in decision making [4]. A group decision-making process can be defined as a decision situation where (1) there are two or more individuals different preferences but the same access to information, each characterized by his/her own perceptions, attitudes, motivations, and personalities; (2) all recognize the existence of a common problem; and (3) all attempt to reach a collective decision [2]. In the group decision-making problems under uncertainty, a decision-maker can provide imprecise or linguistic preference information. For example, when attempting to qualify qualitative phenomenon related to human perception, or precise quantitative information may not be stated, natural language is frequently used rather than numerical values [12]. Hence, the linguistic approach appears to be an important tool for providing a group decision-making framework that incorporates the vagueness and imprecision inherent in AMT justification and selection. An effective means of expressing attributes including process flexibility, product quality, required investment cost, and reduction in setup time, which cannot be assessed using either crisp values or random process, is using linguistic variables or fuzzy numbers [20]. A group of decision makers frequently faces the problem of identifying a solution from a finite set of alternatives. MADM deals with the problem of selecting an alternative from a set of alternatives characterized by multiple attributes. The selected alternative is that which is the most preferred among all the relevant attributes (or predefined objectives). Clearly, the classical MADM methods, both deterministic and random processes, cannot effectively handle group decision-making problems with imprecise and linguistic information, and thus FMADM methods were developed. FMADM methods basically involve two phases before achieving a decision [31]: aggregation and exploitation. The aggregation phase combines the performance ratings for all attributes with respect to each alternative. The exploitation phase ranks the alternatives with respect to the global aggregated performance ratings. Various methods exist for solving these two main phases of FMADM. The literature contains numerous applications of FMADM to different aspects of selection problems with vague data, e.g., robot selection [23], propulsion system selection [28], advanced manufacturing systems selection [7,20,29], etc. Furthermore, there are a great number of FMADM methods related to evaluation, such as manufacturing flexibility [8], design evaluation [25], engineer evaluation [24], and so on.

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In a fuzzy environment, a multiple attribute group decision-making problem considered in this study composes the following elements: Let A = {A1 ,A2 ,. . . ,Am } comprise a finite set of alternatives (courses of AMT), and moreover let there be a finite set of attributes C = {C1 ,C2 ,. . . ,Ck }, where these attributes are classified as subjective attributes {C1 ,C2 ,. . . ,Cs } and objective attributes {Cs+1 ,Cs+2 ,. . . ,Ck }, and a group of n experts E = {E1 ,E2 ,. . . ,En } with each expert Ej ∈ E presenting his/her preference (or opinion) on Ai with respect to Ct . According to the opinions obtained, both direct and indirect approaches can be employed to derive solutions [15,19]. A direct approach obtains a solution directly from the individual opinion. Meanwhile, an indirect approach determines a solution from a group consensus opinion that can be obtained by aggregating the opinions of a group of individuals. This study considers the indirect approach. A general procedure for group decision-making with fuzzy multiple attributes analysis requires three major stages [17]: First, the opinions (or evaluations) from each expert should be unified. The second stage then aggregates unified opinions to form a collective opinion for each alternative. This opinion is usually a fuzzy number or linguistic label and is used to order the alternatives. The third stage involves selecting preferred alternative(s) based on their ranking order. Therefore, a new AMT selection method based on fuzzy multiple attribute group decision-making is developed in the three stages of this procedure. 2.3. Linguistic assessments The linguistic assessment is an approximate method based on linguistic variables. The concept of linguistic variables is extremely useful in dealing with decision situations, which are too complex or ill-defined to be reasonably described using conventional quantitative expressions [39]. A linguistic variable is one whose values are not numbers but rather words or sentences in a natural or artificial language [40]. For example, the opinions of experts regarding subjective or objective attributes for an alternative can be linguistic labels or approximate numerical values, respectively. However, in the real world, the linguistic approach is appropriate for application to many decision situations; restated, while a decision-maker cannot generally specify precise numerical values they can take the form of linguistic variables or fuzzy numbers because (1) a decision should be made to experience time pressure and lack of knowledge or data [35]; (2) numerous attributes are subjective or intangible owing to being unquantifiable in nature [39]; and (3) as for objective attributes, precise quantitative or non-monetary information may not be stated because it is either unavailable or too costly to compute [12]. This approach allows the representation of expert information more directly and adequately [16]. Therefore, the linguistic approach is used in different fields, and decision-making includes numerous approaches based on linguistic information. In applying a linguistic approach to AMT selection, since numerous attributes have been considered in evaluating AMT suitability, these attributes can be identified by considering specific manufacturing system requirements. In general, the attributes are classified as subjective and objective. For each subjective attribute, expert opinions regarding individual alternative can be linguistic terms (labels), which are characterized by trapezoidal fuzzy numbers. For objective attributes, expert opinions are expressed as approximate numerical values characterized with trapezoidal fuzzy numbers. As mentioned above, in order to establish the decision matrix for each expert, a group of decision-makers (or experts) express their opinions (or preferences) for each alternative with respect to each attribute. These opinions can be obtained by direct assignment in utility functions, and can be linguistic terms or approximate numerical values, also seen in many literatures of group decision-making [20, p. 55; 23, p. 269; 28, p. 100]. With respect to each alternative, the performance ratings and importance grade for each subjective attribute should be rated scored on a linguistic term set (or linguistic scale). The strongest assessment is assigned the highest (or lowest) label ‘definitely high’ (or ‘definitely low’) on a linguistic scale. The elements of the term set determine the granularity of the uncertainty. Furthermore, let S = {s0 ,s1 ,. . . ,sT } be a finite and totally ordered term set with an odd cardinal, where the middle term represents ‘average’, i.e., a probability of ‘approximately 0.5’, and the remaining terms are ordered symmetrically around it and exhibit the following properties [15]: 1. 2. 3. 4.

The set is ordered: si  sj if i  j. The negation operator is defined as Neg(si ) = sj such that j = T−i. The maximization operator is Max (si , sj ) = si if si  sj . The minimization operator is Min (si , sj ) = sj if si  sj .

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For example, a linguistic scale S1 comprising seven terms could be represented as follows: S1 = {s0 = DL, s1 = V L, s2 = L , s3 = M, s4 = H, s5 = V H , s6 = D H }, where DL = definitely low, VL = very low, L = low, M = middle, H = high, VH = very high, DH = definitely high. The semantic of the terms of the linguistic scale is provided by fuzzy numbers defined on the interval [0, 1], which are characterized by membership functions. The use of linguistic variables increases the flexibility and reliability of decision-maker evaluations, but complicates the aggregation of the linguistic labels. Generally, two main approaches are used to aggregate and compare linguistic terms in group decision-making: the first approach uses the associated membership functions, while the second calculates linguistic labels directly. Most methods belong to the first approach. Among these methods, Chen and Hwang [3] presented a numerical approximation system that individually transforms linguistic terms according to their corresponding fuzzy numbers. As for the other methods, the introduction of the linguistic ordered weighted averaging (LOWA) method by Herrera et al. [15] opened the way to act by direct computation on labels. The LOWA operators are based on the ordered weighted averaging (OWA) operators defined by Yager [36] and on the convex combination of linguistic labels defined by Delgado et al. [9]. Chuu [8] proposed a modified LOWA operator with a maximum entropy weighting vector for assessing manufacturing flexibility. This study thus applies a new fuzzy fusion method based on linguistic assessments for AMT evaluation. 3. Fusion of fuzzy information In the fuzzy assessment of AMT, fuzzy numbers are very useful in improving information representation and processing in a fuzzy environment. Trapezoidal (or triangular) fuzzy numbers have been used to characterize linguistic labels (or approximate numerical values) used in approximate reasoning. With respect to approximate numerical values, these fuzzy numbers have previously been used in research articles pertaining to application of fuzzy sets to robot selection [23, p. 268], evaluating advanced manufacturing system investments [20, p. 53], and propulsion system selection [28, p. 100]. The trapezoidal fuzzy numbers are chosen for application considering their intuitive representation, ease in computation, and good enough to capture the vagueness of fuzzy assessment [16,20,28]. Let a fuzzy number A be a special fuzzy subset of a universal set X with membership function A (x), which is a continuous mapping from each element x in X to a real number in the interval [0, 1]. This study assumes that a trapezoidal fuzzy number B = (n1 , n2 , n3 , n4 ) is represented by the membership function B (x) given below: ⎧ (x − n 1 )/(n 2 − n 1 ), n 1  x  n 2 , ⎪ ⎨ 1, n 2  x n 3 ,  B (x) = (1) )/(n − n ), n3  x  n4, (x − n ⎪ 4 3 4 ⎩ 0 otherwise with n1  n2 n3  n4 . The x in interval [n2 , n3 ] yields the maximal grade of B (x), i.e., B (x) = 1, which is the most likely value of the evaluation data. Meanwhile, n1 and n4 comprise the lower and upper limits of the available area for the evaluation data, respectively, which are used to reflect the fuzziness of the evaluation data. Therefore, following the decision process for managing multi-granularity linguistic information [14], but considering our particular decision context, i.e., to handle compatibility between approximate numerical values and linguistic labels, this study presents a fusion method of fuzzy information, which is performed in two phases: 1. Making the information uniform. 2. Computing the collective information. They are analyzed in the following subsections. 3.1. Making the information uniform For handling all the fuzzy data, the expert fuzzy assessments must be converted into a basic linguistic scale [14]. Each assessment value is defined as a fuzzy set on the basic linguistic scale. With respect to the assessments made by linguistic labels in the linguistic scale, let S = {s0 ,s1 ,. . . ,sT } and V = {v0 ,v1 ,. . . ,vG } be two linguistic scales, such that

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Table 1 Linguistic variables and fuzzy numbers of basic linguistic scale Linguistic variable

Fuzzy number

v0 : Definitely low (DL) v1 : Extra low (EL) v2 : Very low (VL) v3 : Low (L) v4 : Slightly low (SL) v5 : Middle (M) v6 : Slightly high (SH) v7 : High (H) v8 : Very high (VH) v9 : Extra high (EH) v10 : Definitely high (DH)

(0.0, 0.0, 0.0, 0.1) (0.0, 0.1, 0.1, 0.2) (0.1, 0.2, 0.2, 0.3) (0.2, 0.3, 0.3, 0.4) (0.3, 0.4, 0.4, 0.5) (0.4, 0.5, 0.5, 0.6) (0.5, 0.6, 0.6, 0.7) (0.6, 0.7, 0.7, 0.8) (0.7, 0.8, 0.8, 0.9) (0.8, 0.9, 0.9, 1.0) (0.9, 1.0, 1.0, 1.0)

G T. A transformation function SV is then defined as [14]  SV : S → F(V ),  SV (Si ) = {(u i j , v j )/j ∈ {0, 1, . . . , G}} for si ∈ S, u i j = max min{si (x), v j (x)}, x

(2)

where F(V) denotes the set of fuzzy sets defined in V, which is a basic linguistic scale, and si (x), vj (x) represent the membership functions of the fuzzy sets associated with the labels si and vj , respectively. Example 1. Let V1 = {v0 ,v1 ,. . . ,v10 }denote a basic linguistic scale, as shown in Table 1, and let S1 = {s0 = (0,0,0.1,0.2), s1 = (0.1,0.2,0.3,0.4), s2 = (0.3,0.4,0.6,0.7), s3 = (0.6,0.7,0.8,0.9), s4 = (0.8,0.9,1.0,1.0)} with 11 and five labels, respectively. Therefore, with respect to each linguistic label of S1 , the result of S1V1 is a fuzzy set defined in V1 that can be obtained using Eq. (2):  S1 V1 (s0 ) = {(1, v0 ), (1, v1 ), (0.5, v2 ), (0, v3 ), (0, v4 ), (0, v5 ), (0, v6 ), (0, v7 ), (0, v8 ), (0, v9 ), (0, v10 )},  S1 V1 (s1 ) = {(0, v0 ), (0.5, v1 ), (1, v2 ), (1, v3 ), (0.5, v4 ), (0, v5 ), (0, v6 ), (0, v7 ), (0, v8 ), (0, v9 ), (0, v10 )},  S1 V1 (s2 ) = {(0, v0 ), (0, v1 ), (0, v2 ), (0.5, v3 ), (1, v4 ), (1, v5 ), (1, v6 ), (0.5, v7 ), (0, v8 ), (0, v9 ), (0, v10 )},  S1 V1 (s3 ) = {(0, v0 ), (0, v1 ), (0, v2 ), (0, v3 ), (0, v4 ), (0, v5 ), (0.5, v6 ), (1, v7 ), (1, v8 ), (0.5, v9 ), (0, v10 )},  S1 V1 (s4 ) = {(0, v0 ), (0, v1 ), (0, v2 ), (0, v3 ), (0, v4 ), (0, v5 ), (0, v6 ), (0, v7 ), (0.5, v8 ), (1, v9 ), (1, v10 )}. As mentioned above, all the information sources using the same scale ([0,1]) are considered. Regarding the fuzzy assessments made by approximate numerical values, the transformation function also appropriately converted the standardized fuzzy assessments, the ranges of which belong to [0, 1], into a basic linguistic scale. The max–min operation has been used in SV , because it is a classical tool for setting the degree of matching between fuzzy sets [40]. The following subsection presents how to obtain the collective assessments for each attribute (or expert). 3.2. Computing the collective information The converted information (performance rating) provided by an expert for each alternative with respect to attribute is defined as the fuzzy set on the basic linguistic scale V. The collective performance rating of an alternative is then obtained by aggregating these fuzzy sets. The collective performance rating is also a new fuzzy set defined on V. This paper considers the MEOWA operator as the aggregation operator. An MEOWA operator is used because it is not only feasible but also effective. It is elicited as follows. The OWA operators introduced by Yager [36] provide a family of aggregation operators lying between the ‘and’ and the ‘or’, and a unified framework for decision-making under uncertainty, in which different decision criteria

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such as maximax (optimistic), maximin (pessimistic), equally likely (Laplace), and Hurwicz criteria are characterized by different OWA operator weights [33]. With respect to the OWA operator weights, Yager [36] also provided two measures, namely ‘orness’ and ‘dispersion (or entropy)’. The orness is a value that lies in [0, 1], and measures the degree to which the aggregation resembles an ‘or’ operation, and can be considered a gauge of decision-maker optimism. The more closely the orness of an OWA operator approaches the ‘or’ operator, the more the optimistic decision-maker is about obtaining the best solution. The dispersion measures the degree to which all the aggregates are equally used. In the framework of multiple attribute group decision-making under uncertainty, the OWA operators can be provided for aggregating the attributes (experts) associated with some linguistic fuzzy quantifiers [19], such as ‘as many as possible’, ‘most’, ‘average’, ‘almost all’, ‘at least half’, etc., used to determine the weights. In group decision-making, linguistic fuzzy quantifiers are used to indicate a fusion strategy for guiding the process of aggregating expert opinions. OWA operators have been applied numerous fields, including neural networks, decision-making, data base systems, learning systems and fuzzy logic controllers, communication networks, and so on, as described by Yager [37]. To determine OWA operator weights, O’Hagan [27] developed a maximum entropy approach, which formulates the problem as a constraint nonlinear optimization model with a predefined degree of orness as its constraint and the entropy as its objective function. The resultant weights and OWA operators are termed the maximum entropy weights and MEOWA operators, respectively. Filev and Yager [10] examined the analytical properties of MEOWA operators and proposed a two-step process for obtaining the maximum entropy weights that generate some prescribed orness without having to solve the constraint nonlinear optimization problem. Furthermore, Fuller and Majlender [11] developed a minimum variance method, which requires the solution of the quadratic programming problem to obtain the minimal variability OWA operator weights under a given degree of orness. Wang and Parkan [34] presented a minimax disparity approach, which is formulated as a linear programming model with the maximum disparity between two adjacent weights under a given level of orness. Recently, Wang et al. [33] suggested the least squares and chi-square methods, which both are constrained nonlinear optimization problem to produce as equally important OWA operator weights as possible for a given degree of orness, could be used to obtain the least squares weights and chi-square weights, respectively. However, all the above methods imply the use of more information from all attributes. Wang et al. [33] also showed that the OWA operator weights determined via these methods differ slightly from one to another, but not significantly. In practice, Chuu [8] also proposed an FMADM model based on MEOWA operators for evaluating manufacturing flexibility. These studies thus showed that using the maximum entropy method to determine MEOWA operator weights is both feasible and effective. An MEOWA operator of dimension n is a mapping  : R n → R, which has an associated maximum entropy weighting vector W ∗ = [w1∗ , w2∗ , . . . , wn∗ ] with wi∗ ∈ [0, 1] and such that n  (a1 , a2 , . . . , an ) = w ∗j b j ,

n

∗ i=1 wi = 1

(3)

j=1

where bj is the jth largest element in the collection {a1 ,a2 ,. . . ,an }. An algorithm for calculating W* is as follows [6,10,19,36]: Step 1: Determine a non-decreasing proportional linguistic fuzzy quantifier Q for representing the fuzzy majority over decision-makers or attributes, as follows: (1) Some non-decreasing proportional linguistic fuzzy quantifiers are typified by terms ‘most’, ‘at least half’, and ‘as many as possible’, the respective parameters (a, b) of which are (0.3, 0.8), (0, 0.5), and (0.5, 1), respectively, is defined as 0 if r < a Q(r ) = (r − a)/(b − a) if ar b (4) 1 if r > b with a, b, r ∈ [0, 1]. (2) An alternative non-decreasing proportional linguistic fuzzy quantifier ‘most of’ is defined as follows: Q(r ) = r 1/2 with r ∈ [0, 1].

(5)

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Step 2: Compute the weighting vector W, W (i) = Q(i/n) − Q((i − 1)/n) for i = 1, 2, . . . , n. Step 3: Compute the orness value , n

  = (n − i)W (i) (n − 1).

(6)

(7)

i=1

Step 4: Compute the W*, using the two-step process. 4-1: Find a positive solution h* of the algebraic equation, n 

((n − i)/(n − 1) − )h (n−i) = 0.

(8)

i=1

4-2: Obtain W* from the following equation, using * = (n−1) ln h*, ∗

e ×((n−i)/(n−1)) for i = 1, 2, . . . , n. W ∗ (i) = n ∗ ×((n− j)/(n−1)) j=1 e

(9)

Example 2. Consider a situation with a collection involving four elements {a1 = 0.1, a2 = 0.2, a1 = 0.3, a2 = 0.4}, and where the MEOWA operator is applied, in which case using a linguistic fuzzy quantifier ‘most’ with the pair (0.3, 0.8), the following aggregation value can be obtained: An algorithm for calculating W* is as follows: W = [0, 0.4, 0.5, 0.1], in which W (2) = Q(2/4) − Q(1/4) = ((0.5 − 0.3)/(0.8 − 0.3)) − 0 = 0.4,  = ((4 − 1) × 0 + (4 − 2) × 0.4 + (4 − 3) × 0.5 + (4 − 4) × 0.1)/(4 − 1) = 0.4333. W* = [0.1932, 0.2269, 0.2666, 0.3133], in which the following algebraic equation can be obtained: (1 − 0.433) × h 3 + (2/3 − 0.433) × h 2 + (1/3 − 0.433) × h = 0. Its positive solution h* = 0.8511 and its associated parameter * = (4−1) ln 0.8511 = −0.4835, e−0.4835×((4−2)/(4−1)) W ∗ (2) =  −0.4835×((4− j)/(4−1)) , je = 0.2269.

j = 1, 2, 3, 4,

Then we have ‘most’ (0.1, 0.2, 0.3, 0.4) = W ∗ • B T = [0.1932, 0.2269, 0.2666, 0.3133] • [0.4, 0.3, 0.2, 0.1]T = 0.2298. 4. Evaluating the suitability of AMT This section presents a new AMT selection method using fuzzy multiple attribute analysis and group decision-making to overcome above problems. This method enables the experts’ fuzzy assessments with the linguistic and numerical scales can be considered in the aggregation process. An algorithm is developed in three main stages, as follows: 1. Qualitative assessment stage. 2. Quantitative evaluation stage. 3. Selection stage.

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The first stage assesses the qualitative effects of AMT with respect to subjective attributes. With respect to objective attributes, the second stage evaluates the quantitative effects of AMT. The last stage aggregates the assessments, ranks the alternatives, and make a decision. A group of n experts is responsible for evaluating the suitability of alternatives. To establish the assessment data for each expert, experts express their opinions regarding each alternative with respect to each attribute. This can be conducted using questionnaires, which are used for soliciting expert opinions regarding alternatives. 4.1. Qualitative assessment stage This stage aims to obtain the weighted performance ratings versus subjective attributes, and then convert them to yield the fuzzy assessment vectors. In the fuzzy linguistic assessment, let S be an appropriate linguistic scale chosen by experts to be used for the qualitative assessment versus subjective attributes. For alternative Ai , the symbols Wjt and Rijt are linguistic labels belonging to S, and are used to denote the importance grade and related performance rating for subjective attribute Ct , respectively, according to the assessment data of expert Ej (i = 1,2,. . . ,m; j = 1,2,. . . ,n; t = 1,2,. . . ,s). This work applies a convex combination of linguistic labels through direct computation; restated, the independence of the type of membership functions being used [9,15]. Consequently, the weighted rating Xijt is also a linguistic label belonging to S, which is calculated by Eqs. (10) and (11), as follows: X i jt = C 2 (W jt , Ri jt ) = w1 ⊗ sa ⊕ (1 − w1 ) ⊗ sb = sc , sb , sa ∈ S(ba) for i = 1, 2, . . . , m, j = 1, 2, . . . , n, t = 1, 2, . . . , s,

(10)

such that c = Min(T, a + r ound(w1 × (b − a))),

(11)

where ⊗ denotes the general product of a linguistic label by a positive real number, ⊕ represents the general addition of linguistic labels, and ‘round’ is a usual round operation. For handling the fuzzy information, all the linguistic weighted ratings are transformed into their corresponding fuzzy numbers with linguistic scale S, after which these corresponding fuzzy numbers are converted into a basic linguistic scale. Therefore, each weighted rating can be defined as a fuzzy set on the basic linguistic scale. This paper considers a basic linguistic scale V1 with 11 labels v0 , v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 , v9 , v10 , which are treated as trapezoidal fuzzy numbers with membership function (x), as presented in Table 1. Finally, using Eq. (2), the corresponding fuzzy numbers of Xijt are converted into the basic linguistic scale V1 . Thus the fuzzy assessment vector on V1 , F(Xijt ) can be formed as follows: F(X i jt ) = (u(X i jt , v0 ), u(X i jt , v1 ), . . . , u(X i jt , v10 )) for i = 1, 2, . . . , m, j = 1, 2, . . . , n, t = 1, 2, . . . , s.

(12)

4.2. Quantitative evaluation stage During the second stage, the objective attributes can be classified into benefit and cost attributes. Regarding objective attributes, since precise quantitative information may be unavailable, and since evaluating this information is expensive, quantitative effects of AMT are evaluated by management using approximate numerical values, and then identified by experts. In this study these estimated performance ratings are generally expressed in sentence form as ‘approximately between’. For instance, an expert’s estimate is ‘approximately between 6 and 6.5’ with ‘5 and 7 are the lower and upper limits of the available area for the evaluation data’, it can be represented by a trapezoidal fuzzy number as (5, 6, 6.5, 7). To ensure compatibility among the various numerical scales, all the estimated values must be converted into a comparable scale. This stage attempts to obtain the weighted ratings versus objective attributes, and then convert them to yield the fuzzy assessment vectors. During this stage, the conversion is performed by linear scale transformation as defined by Hsu and Chen [18], since this transformation preserves the property that the ranges of standardized trapezoidal fuzzy numbers belong to [0, 1]. Let a trapezoidal fuzzy number Wjt = (ajt , bjt , cjt , djt ) be used to denote the importance grade for objective attribute Ct assigned by each expert Ej and let Hit = (eit , git , hit , lit ) represent a positive trapezoidal fuzzy number representing the

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estimated performance rating for an alternative Ai with respect to an objective attribute Ct , where 0eit  git hit  lit (i = 1,2,. . . ,m; j = 1,2,. . . ,n; t = s+1, s+2,. . . ,k). Then the standardized performance rating Rit = (oit , pit , qit , rit ) is also a trapezoidal fuzzy number, which is calculated by Eqs. (13) and (14), respectively, and 0oit pit  qit rit 1, Rit = (eit /lt+ , git /lt+ , h it /lt+ , lit /lt+ ) if t ∈ B,

(13)

Rit = (et− /eit , et− /git , et− / h it , et− /lit ) if t ∈ C,

(14)

where B and C denote the sets of benefit and cost attributes, respectively, and lt + = maxi lit , et − = mini eit . Thus, the weighted rating can be calculated using X i jt = W jt ⊗ Rit = (Ui jt , Vi jt , Yi jt , Z i jt ) for i = 1, 2, . . . , m, j = 1, 2, . . . , n, t = s + 1, s + 2, . . . , k,

(15)

where operator ⊗ denotes the fuzzy multiplication operator, and Uijt = ajt oit , Vijt = bjt pit , Yijt = cjt qit , Zijt = djt rit . The fuzzy multiplication of trapezoidal fuzzy numbers is also a trapezoidal fuzzy number [21]. Therefore, Xijt is a trapezoidal fuzzy number defined by Eq. (15), as described by Karsak and Tolga [20]. These fuzzy numbers can then be converted into a basic linguistic scale. Using Eq. (2), the weighted rating Xijt is converted into the basic linguistic scale V1 . Thus the fuzzy assessment vector on V1 , F(Xijt ) can be represented as follows: F(X i jt ) = (u(X i jt , v0 ), u(X i jt , v1 ), . . . , u(X i jt , v10 )) for i = 1, 2, . . . , m, j = 1, 2, . . . , n, t = s + 1, s + 2, . . . , k.

(16)

4.3. Selection stage This stage proposed an indirect approach for a group of experts. Since the aggregation is based upon individual attribute, the maximum entropy weighting vector obtained from a linguistic fuzzy quantifier represents the fuzzy majority over the n experts. The linguistic fuzzy quantifier can be used to measure the degree of decision-maker optimism. One means of modeling this aspect is to consider the existence of a manager (or moderator) that assigns a linguistic fuzzy quantifier to each attribute. This stage establishes an algorithm for combining the opinions of a group of experts to form a group consensus opinion. This algorithm uses MEOWA operators to aggregate the parameter assessed by each expert, and aggregates the results to yield the ranking order, and uses interactive decision analysis for selecting the appropriate AMT. The aggregation algorithm for a group of experts is presented as follows: (1) Aggregate F(Xijt ) to yield the fuzzy assessment vector (F(XA(it) )). The aggregated parameters obtained from the assessment data of n experts can be obtained by X A(it) (v y ) =  Q 1 (u(X i1t , v y ), u(X i2t , v y ), . . . , u(X int , v y )) for i = 1, 2, . . . , m, t = 1, 2, . . . , k, y = 0, 1, . . . , 10,

(17)

where  Q 1 denotes the MEOWA operator with the maximum entropy weighting vector W1∗ , obtained from a linguistic fuzzy quantifier Q1 , which represents the fuzzy majority over the n experts. Then, the fuzzy assessment vector on V1 under attribute Ct , F(XA(it) ) is defined as F(X A(it) ) = (u(X A(it) , v0 ), u(X A(it) , v1 ), . . . , u(X A(it) , v10 )) for i = 1, 2, . . . , m, t = 1, 2, . . . , k.

(18)

(2) Aggregate F(XA(it) ) to yield the fuzzy suitability vector (F(XA(i) )). Using the concept of fuzzy majority over the attributes specified by a linguistic fuzzy quantifier Q2 , and using the MEOWA operator associated with W2∗ , yields the aggregated parameters for alternative Ai , as follows: X A(i) (v y ) =  Q 2 (u(X A(i1) , v y ), u(X A(i2) , v y ), . . . , u(X A(ik) , v y )) for i = 1, 2, . . . , m, y = 0, 1, . . . , 10.

(19)

Then, the fuzzy suitability vector on V1 for Ai , F(XA(i) ) is defined as F(X A(i) ) = (X A(i) (v1 ), X A(i) (v2 ), . . . , X A(i) (v10 )) for i = 1, 2, . . . , m.

(20)

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(3) Defuzzify F(XA(i) ) to yield the ranking value SAi . To rank the alternatives involved in the problem, it is necessary to defuzzify all the fuzzy suitability vectors. In this research, Eq. (21) defines the defuzzification employed by the centroid method. This method is used because it is intuitive and easy to implement.

b b x(x) dx (x) dx, (21) a

a

where ‘a’ and ‘b’ are the lower and upper limits of the integral, respectively. This work has its centroids G(v0 ) = 0.0333, G(v1 ) = 0.1, G(v2 ) = 0.2, G(v3 ) = 0.3, G(v4 ) = 0.4, G(v5 ) = 0.5, G(v6 ) = 0.6, G(v7 ) = 0.7, G(v8 ) = 0.8, G(v9 ) = 0.9, and G(v10 ) = 0.9667 as the center of mass of v0 , v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 , v9 , and v10 , respectively. Using Eq. (22), the SAi for alternative Ai is obtained as follows:  10 10   S Ai = G(v y ) × X A(i) (v y ) G(v y ) for i = 1, 2, . . . , m. (22) y=0

y=0

Clearly, SAi can be considered as the suitability of alternative Ai . The ranking order and most suitable alternative(s) are determined based on ranking values. (4) Interactive decision analysis. It analyzes that the evaluation results obtained in the before stages is accurate and reliable enough to make a consistent decision. According to this analysis, the process has to go back to the initial stages in order to gather additional information of the problem, or has to accept the evaluation results in order to accomplish the decision-making process. The proposed approach presented above includes the following steps: Step 1. Form a committee of experts (or decision-makers), and identify the alternatives available for consideration. Step 2. Identify the selection attributes (subjective or objective) with types (cost or benefit) of them. Step 3. Determine an appropriate linguistic scale chosen by experts using qualitative assessment versus subjective attributes, and identify the appropriate numerical scales using the quantitative evaluation versus objective attributes. Qualitative assessment stage (with respect to subjective attributes): Step 4. Collect expert opinions (performance rating and importance grade) for each alternative. Step 5. Aggregate the performance rating and importance grade to obtain a linguistic weighted rating (Xijt ). Step 6. Transform Xijt into its corresponding fuzzy number by using an appropriate linguistic scale as determined in Step 3. Step 7. Convert the corresponding fuzzy number of Xijt to yield the fuzzy assessment vector (F(Xijt )). Quantitative evaluation stage (with respect to objective attributes): Step 8. Identify expert opinions (performance rating and importance grade) for each alternative. Step 9. Standardize the performance ratings. Step 10. Aggregate the standardized performance rating and importance grade to obtain the numerical weighted rating (Xijt ). Step 11. Convert the corresponding fuzzy number of each Xijt to obtain the fuzzy assessment vector (F(Xijt )). Selection stage: Step 12. Aggregate F(Xijt ) to yield the fuzzy assessment vector (F(XA(it) )) for alternative Ai with respect to attribute Ct . Step 13. Aggregate F(XA(it) ) to yield the fuzzy suitability vector (F(XA(i) )) for alternative Ai . Step 14. Defuzzify F(XA(i) ) to yield the ranking value (SAi ) for alternative Ai , and then rank the alternatives according to SAi , and select the alternative with the maximum SAi as the best. Step 15. Interactive decision analysis. According to a group of experts analyze the evaluation results, the process has to go back the initial stages or has to accept the evaluation. 5. Case study In this section, the application of the proposed selection method to the case of a leading Taiwanese firm in the bicycle industry is discussed. For reasons of confidentiality, the name of the firm is not revealed. This case originated from a feasibility evaluation of Optional Operation System for the bicycle industry, and the data were taken from a study entitled ‘Fuzzy multi-attribute decision-making for evaluating manufacturing flexibility’ [8].

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Table 2 The FMS selection attributes Subjective attribute

Type of assessment

C1 : Process flexibility C2 : Product quality C3 : Learning C4 : Exposure to labor unrest Objective attribute

C5 : Required floor space

Important grade

Performance rating

Linguistic Linguistic Linguistic Linguistic

Linguistic Linguistic Linguistic Linguistic

Type of assessment

(ft2×1000)

Type of attribute

Important grade

Performance rating

Linguistic

Fuzzy (as approximately between) Fuzzy (as approximately between) Fuzzy (as approximately between) Fuzzy (as approximately between)

C6 : Capacity (unit×1000)

Linguistic

C7 : Lead time (h)

Linguistic

C8 : Purchase cost ($×10,000)

Linguistic

Cost Benefit Cost Cost

Since increasing customization, a few years ago the case firm decided that it needed a flexible manufacturing system (FMS) that would allow a customized bike, designed according to the requirements of the customer, to be delivered within a week. After performing task analysis, it was confirmed that the new system would produce customized mountain and road racing bikes. Each bike consists of 11 subsystems including a frame, suspension fork, derailleur shifters, brokers, hubs and rims, tires, pedals, handle bar, stem, saddle, and seat post. Furthermore, each subsystem includes several models among which customers can select. Following preliminary screening, three competing FMS alternatives, A1 , A2 , and A3 , are identified that are capable of performing this production task. For evaluating the suitability of FMS, the alternatives considered in this case study are evaluated by a group of experts, E1 , E2 , and E3 regarding each attribute. The effects of adopting a FMS include both quantitative and qualitative effects. Subjective attributes are used to assess the qualitative effects of FMS, while objective attributes are used to evaluate the quantitative effects of FMS. The prospective FMS buyer can offer to spend at most $1,700,000. Objective attributes have been identified including that desired capacity should be at least 15,000 units per day, required floor space should be at most 10,000 ft2 and lead times should be 12 h or less. In the industry considered, the subjective attributes identified are process flexibility, product quality, learning, and exposure to labor unrest. The selection decision is made based on four objective attributes and four subjective attributes. Table 2 lists the properties of these attributes, including attribute type and assessment type, which are critical to FMS function. Since it is useful to develop a hierarchical structure showing the overall goal, as well as the attributes and alternatives, this hierarchy for the FMS selection problem is shown in Fig. 1. This paper chooses a basic linguistic scale V1 with 11 labels, as listed in Table 1. For convenience, with respect to the alternative A1 , the proposed algorithm can be expressed as follows. 5.1. Qualitative assessment stage calculations (Steps 4–7) For each of the subjective attributes (C1 , C2 , C3 , and C4 ), experts provide their opinions using the linguistic scale S1 , as presented in Table 3. Consequently, Table 4 lists expert linguistic assessments for each alternative. For example, the performance ratings are assessed by a group of experts (E1 , E2 , and E3 ) for alternative A1 with respect to C1 are ‘VL’, ‘L’, and ‘VL’, respectively. Using Eqs. (10) and (11), and letting w1 = 0.5, the weighted ratings for each FMS versus subjective attributes can be obtained, as presented in Table 5. With respect to alternative A1 , expert linguistic assessments for all the subjective attributes can be converted to trapezoidal fuzzy numbers using the linguistic scale S1 ; for example, ‘VL’ can be represented by the trapezoidal fuzzy number (0.1, 0.2, 0.2, 0.3), and these fuzzy numbers are then transformed into

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Overall goal

Select the best FMS alternative

Attributes Objective

Subjective

Required floor space

Process flexibility

Capacity

Product quality

Lead time

Learning

Investment cost

Exposure to labor unrest

Alternatives FMS1

FMS2

FMS3

Fig. 1. Decision hierarchy of FMS evaluation problem.

Table 3 Linguistic variables of performance rating and importance grade Seven ranks of performance rating

Fuzzy number

Seven ranks of importance grade

Fuzzy number

s0 : Definitely low (DL) s1 : Very low (VL) s2 : Low (L) s3 : Middle (M) s4 : High (H) s5 : Very high (VH) s6 : Definitely high (DH)

(0, 0, 0.1, 0.2) (0.1, 0.2, 0.2, 0.3) (0.2, 0.3, 0.4, 0.5) (0.4, 0.5, 0.5, 0.6) (0.5, 0.6, 0.7, 0.8) (0.7, 0.8, 0.8, 0.9) (0.8, 0.9, 1.0, 1.0)

s0 : Definitely low (DL) s1 : Very low (VL) s2 : Low (L) s3 : Middle (M) s4 : High (H) s5 : Very high (VH) s6 : Definitely high (DH)

(0, 0, 0.1, 0.2) (0.1, 0.2, 0.2, 0.3) (0.2, 0.3, 0.4, 0.5) (0.4, 0.5, 0.5, 0.6) (0.5, 0.6, 0.7, 0.8) (0.7, 0.8, 0.8, 0.9) (0.8, 0.9, 1.0, 1.0)

Table 4 The importance grades and performance ratings evaluated by three experts for three alternatives Subjective attribute

Importance grade

Performance rating E1

C1 C2 C3 C4 Objective attribute

C5 C6 C7 C8

E2

E3

E1

E2

E3

A1

A2

A3

A1

A2

A3

A1

A2

A3

DH VH M VL

VH DH VH M

DH DH M M

VL L VL DL

H M M H

VH DH DH VH

L VL DL VL

M H L M

VH VH DH DH

VL DL L DL

H L M H

DH VH DH DH

Importance grade

Performance rating

E1

E2

E3

A1

A2

A3

M M VH DH

VH H DH VH

H M VH VH

(5, 6, 6.5, 7) (16, 18, 19, 20) (9.5, 10, 10, 11) (90, 100, 110, 120)

(7.5, 8, 8, 9) (24, 25, 25, 26) (4, 5, 5.5, 6.5) (130, 140, 140, 145)

(7, 8.5, 9, 9.5) (29, 31, 31, 32) (2.5, 3, 3.5, 4) (145, 150, 155, 160)

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Table 5 Expert weighted ratings for three alternatives under each subjective attribute Subjective attribute

C1 C2 C3 C4

E1

E2

E3

A1

A2

A3

A1

A2

A3

A1

A2

A3

H H L VL

VH H M M

DH DH VH M

H H M L

H VH H M

VH DH DH VH

H M M L

VH H M H

DH DH VH VH

Table 6 The standardized performance ratings for three alternatives under each objective attribute Objective attribute

A1

A2

A3

C5 C6 C7 C8

(0.7143, 0.7692, 0.8333, 1.0) (0.5, 0.5625, 0.5938, 0.625) (0.2272, 0.25, 0.25, 0.2632) (0.75, 0.8182, 0.9, 1.0)

(0.5556, 0.625, 0.625, 0.6667) (0.75, 0.7813, 0.7813, 0.8125) (0.3846, 0.4545, 0.5, 0.625) (0.6207, 0.6429, 0.6429, 0.6923)

(0.5263, 0.5556, 0.5882, 0.7143) (0.9063, 0.9688, 0.9688, 1.0) (0.625, 0.7143, 0.8333, 1.0) (0.5625, 0.5806, 0.6, 0.6207)

fuzzy assessment vectors (F(Xijt )) on the basic linguistic scale V1 by Eqs. (2) and (12), respectively. The results are obtained as follows: F(X 111 ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0), F(X 113 ) = (0, 0, 0.5, 1, 1, 0.5, 0, 0, 0, 0, 0, ),

F(X 112 ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0), F(X 114 ) = (0, 0.5, 1, 0.5, 0, 0, 0, 0, 0, 0, 0),

F(X 121 ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0),

F(X 122 ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0),

F(X 123 ) = (0, 0, 0, 0, 0.5, 1, 0.5, 0, 0, 0, 0),

F(X 124 ) = (0, 0, 0.5, 1, 1, 0.5, 0, 0, 0, 0, 0),

F(X 131 ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0),

F(X 132 ) = (0, 0, 0, 0, 0.5, 1, 0.5, 0, 0, 0, 0),

F(X 133 ) = (0, 0, 0, 0, 0.5, 1, 0.5, 0, 0, 0, 0),

F(X 134 ) = (0, 0, 0.5, 1, 1, 0.5, 0, 0, 0, 0, 0).

5.2. Quantitative evaluation stage calculations (Steps 8–11) With respect to each of the objective attributes (C5 , C6 , C7 , and C8 ), based on the evaluation information provided by management, the performance rating and important grade for each FMS are identified by a group of experts, as listed in Table 4. Using Eqs. (13) and (14), the standardized performance ratings for each alternative versus the objective attributes are listed in Table 6. For example, for alternative A1 with respect to C5 , the performance rating of (5, 6, 6.5, 7) can be transformed into the standardized performance rating of (0.7143, 0.7692, 0.8333, 1.0) using Eq. (14). Using Eq. (15), the numerical weighted rating X ijt (i = 1,2,3, j = 1,2,3, t = 5,6,7,8) can be listed in Table 7. Also as for alternative A1 , using Eqs. (2) and (16), respectively, expert fuzzy assessments for all the objective attributes are converted into fuzzy assessment vectors (F(Xijt )) on V1 . The following results are thus obtained: F(X 115 ) = (0, 0, 0.0719, 0.3425, 1, 0.7060, 0.353, 0, 0, 0, 0), F(X 116 ) = (0, 0, 0.1031, 0.9826, 0.4211, 0, 0, 0, 0, 0, 0), F(X 117 ) = (0, 0.2908, 1, 0.2695, 0, 0, 0, 0, 0, 0, 0), F(X 118 ) = (0, 0, 0, 0, 0, 0, 0.423, 0.846, 1, 1, 0.5), F(X 125 ) = (0, 0, 0, 0, 0, 0.4643, 0.9285, 0.8982, 0.5988, 0.2994, 0), F(X 126 ) = (0, 0, 0.2667, 0.8, 1, 0.5426, 0, 0, 0, 0, 0), F(X 127 ) = (0, 0.1271, 0.8254, 0.5583, 0, 0, 0, 0, 0, 0, 0), F(X 128 ) = (0, 0, 0, 0, 0, 0.3267, 0.7622, 1, 0.7143, 0.3571, 0),

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Table 7 Expert weighted ratings for three alternatives under each objective attribute A1

A2

A3

E1 C5 C6 C7 C8

(0.2857, 0.3846, 0.4167, 0.6) (0.2, 0.2813, 0.2969, 0.375) (0.159, 0.2, 0.2, 0.2369) (0.6, 0.7364, 0.9, 1.0)

(0.2222, 0.3125, 0.3125, 0.4) (0.3, 0.3907, 0.3907, 0.4875) (0.2692, 0.3636, 0.4, 0.5625) (0.4966, 0.5786, 0.6429, 0.6923)

(0.2105, 0.2778, 0.2941, 0.4286) (0.3625, 0.4844, 0.4844, 0.6) (0.4375, 0.5714, 0.6667, 0.9) (0.45, 0.5225, 0.6, 0.6207)

E2 C5 C6 C7 C8

(0.5, 0.6125, 0.6666, 0.9) (0.25, 0.3375, 0.4175, 0.5) (0.1818, 0.225, 0.25, 0.2632) (0.525, 0.6546, 0.72, 0.9)

(0.3889, 0.5, 0.5, 0.6) (0.3, 0.4688, 0.547, 0.65) (0.3077, 0.4091, 0.5, 0.625) (0.4345, 0.5143, 0.5143, 0.6231)

(0.3684, 0.4445, 0.4706, 0.6429) (0.4532, 0.5813, 0.6782, 0.8) (0.5, 0.6429, 0.8333, 1.0) (0.3938, 0.4645, 0.48, 0.5586)

E3 C5 C6 C7 C8

(0.3572, 0.4615, 0.5833, 0.8) (0.2, 0.2813, 0.2969, 0.375) (0.1136, 0.15, 0.175, 0.2106) (0.525, 0.6546, 0.72, 0.9)

(0.2778, 0.375, 0.4375, 0.5334) (0.3, 0.3907, 0.3907, 0.4875) (0.1923, 0.2727, 0.35, 0.5) (0.4345, 0.5143, 0.5143, 0.6231)

(0.2632, 0.3334, 0.4117, 0.5714) (0.3625, 0.4844, 0.4844, 0.6) (0.3125, 0.4286, 0.5833, 0.8) (0.3938, 0.4645, 0.48, 0.5586)

F(X 135 ) = (0, 0, 0, 0.2095, 0.699, 1, 0.9473, 0.6315, 0.3158, 0, 0), F(X 136 ) = (0, 0, 0.5516, 0.9826, 0.4211, 0, 0, 0, 0, 0, 0), F(X 137 ) = (0, 0.6634, 0.8156, 0.0782, 0, 0, 0, 0, 0, 0, 0), F(X 138 ) = (0, 0, 0, 0, 0, 0.3267, 0.7622, 1, 0.7143, 0.3571, 0). 5.3. Selection stage calculations (Steps 12–14) In this stage, all the fuzzy assessment vectors for a group of experts are aggregated to form a group consensus opinion for each attribute as follows: (1) Aggregate F(Xijt ) to yield the fuzzy assessment vector (F(XA(it) )). Using Eqs. (17) and (18), respectively, the manager (moderator) of the decision problem assigns a linguistic fuzzy quantifier ‘most’ to the corresponding experts; i.e., the MEOWA operator  Q 1 guided by ‘most’ with its parameters (0.3, 0.8), and the algorithm for calculating the maximum entropy weighting vector yields the weighting vector W1 , orness value 1 , and maximum entropy weighting vector W1∗ , as follows: W1 = [0.0667, 0.6667, 0.2667], 1 = 0.4, W1∗ = [0.2384, 0.3233, 0.4384]. The following results are thus obtained: F(X A(11) ) = (0, 0, 0, 0, 0, 0.5, 1, 1, 0.5, 0, 0), F(X A(12) ) = (0, 0, 0, 0, 0.1192, 0.6192, 0.7808, 0.5616, 0.2808, 0, 0), F(X A(13) ) = (0, 0, 0.1192, 0.2384, 0.6192, 0.7808, 0.2808, 0, 0, 0, 0), F(X A(14) ) = (0, 0.1192, 0.4, 0.4, 0.4, 0.4, 0.1192, 0, 0, 0, 0), F(X A(15) ) = (0, 0, 0.0171, 0.1494, 0.4643, 0.6701, 0.6807, 0.4182, 0.2448, 0.0714, 0), F(X A(16) ) = (0, 0, 0.2629, 0.9063, 0.5591, 0.1293, 0, 0, 0, 0, 0), F(X A(17) ) = (0, 0.3079, 0.8627, 0.2545, 0, 0, 0, 0, 0, 0, 0), F(X A(18) ) = (0, 0, 0, 0, 0, 0.1835, 0.6135, 0.9325, 0.7824, 0.5103, 0.1192),

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601

where, for example, the value XA(12) (v5 ) is obtained by Eq. (17): X A(12) (v5 ) = ‘most’ (0.5, 0.5, 1) = 0.6192. (2) Aggregate F(XA(it) ) to yield the fuzzy suitability vector (F(XA(i) )). Using Eqs. (19) and (20), respectively, the MEOWA operator  Q2 guided by ‘most’ with the pair (0.3, 0.8), and the algorithm for calculating W2∗ yields W2 = [0, 0, 0.15, 0.25, 0.25, 0.25, 0.1, 0], 2 = 0.4429, and W2∗ =[0.0941, 0.1016, 0.1097, 0.1185, 0.1279, 0.1381, 0.1491, 0.1610]. The fuzzy suitability vector for alternative A1 thus is obtained as follows: F(X A(1) ) = (0, 0.0411, 0.167, 0.2012, 0.2287, 0.3645, 0.3732, 0.3, 0.1843, 0.0553, 0.0112), where, for example, the value XA(1) (v5 ) is obtained by Eq. (19): X A(1) (v5 ) = ‘most’ (0.5, 0.6192, 0.7808, 0.4, 0.6701, 0.1293, 0, 0.1835) = 0.3645. (3) Defuzzify F(XA(i) ) to yield the ranking value SAi . Using Eq. (22), the ranking value SA1 of alternative A1 can be obtained: S A1 = 0.1843. Similarly, SA2 = 0.1814 and SA3 = 0.2332. For a group of experts, based on ranking value, the ranking order of three alternatives is given as A3  A1  A2 . With respect to the detailed analysis of evaluation results such as competing FMS alternatives, effects of FMS, properties of attributes, computational process, and so on, the decision-making process will be completed if experts accept the evaluation results. Otherwise, experts can modify their opinions step by step through the collection of additional information, or modify the linguistic fuzzy quantifier until a consistent decision is obtained. After the detailed decision analysis of this case study, the group of experts accepts that the best alternative is A3 , while A1 and A2 are ranked second and third, respectively. 6. Conclusion AMT selection is important for improving manufacturing system competitiveness. This study first identified two groups of attributes, and then classified them as either subjective or objective. A fuzzy multiple attributes and group decision-making scenario was modeled to solve the AMT evaluation problem. The proposed method applied in the group decision-making is more suitable for solving manufacturing technology evaluation problems involving subjective and imprecise information. The model described in this study involves group decision-making, by which the assessments of decision-makers are more objective and unbiased than those individually assessed. In the proposed method, we also present a fusion approach of fuzzy information. According to decision-maker attitude, a linguistic fuzzy quantifier chosen by the manager of the decision problem is used in MEOWA operators. The proposed method enables the group to incorporate and aggregate fuzzy information provided by multiple decision-makers. A case study of FMS selection has been conducted to exemplify the feasibility of the proposed method. Acknowledgements This research is partially supported by Grant no. NSE 96-2416-H253-001 from the National Science Council of the Republic of China. The author thanks the anonymous referees for their valuable comments that have led to an improved version of this paper. References [1] M.G. Abdel-Kader, D. Dugdale, Evaluating investments in advanced manufacturing technology: a fuzzy set theory approach, British Accounting Rev. 33 (2001) 455–489. [2] T.X. Bui, Co-oP: A Group Decision Support System for Cooperative Multiple Criteria Group Decision Making, Springer, Berlin, 1987. [3] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision-Making: Methods and Applications, Springer, New York, 1992. [4] Z. Chen, D. Ben-Arieh, On the fusion of multi-granularity linguistic label sets in group decision making, Comput. Indust. Eng. 51 (2006) 526–541.

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